Graph Theory and Additive Combinatorics

This course examines classical and modern developments in graph theory and additive combinatorics, with a focus on topics and themes that connect the two subjects. The course also introduces students to current research topics and open problems. This course was previously numbered 18.217.

• Lecture 1: A bridge between graph theory and additive combinatorics
• Lecture 2: Forbidding a Subgraph I: Mantel’s Theorem and Turán’s Theorem
• Lecture 3: Forbidding a Subgraph II: Complete Bipartite Subgraph
• Lecture 4: Forbidding a Subgraph III: Algebraic Constructions
• Lecture 5: Forbidding a Subgraph IV: Dependent Random Choice
• Lecture 6: Szemerédi’s Graph Regularity Lemma I: Statement and Proof
• Lecture 7: Szemerédi’s Graph Regularity Lemma II: Triangle Removal Lemma
• Lecture 8: Szemerédi’s Graph Regularity Lemma III: Further Applications
• Lecture 9: Szemerédi’s Graph Regularity Lemma IV: Induced Removal Lemma
• Lecture 10: Szemerédi’s Graph Regularity Lemma V: Hypergraph Removal and Spectral Proof
• Lecture 11: Pseudorandom Graphs I: Quasirandomness
• Lecture 12: Pseudorandom Graphs II: Second Eigenvalue
• Lecture 13: Sparse Regularity and the Green-Tao Theorem
• Lecture 14: Graph Limits I: Introduction
• Lecture 15: Graph Limits II: Regularity and Counting
• Lecture 16: Graph Limits III: Compactness and Applications
• Lecture 17: Graph Limits IV: Inequalities between Subgraph Densities
• Lecture 18: Roth’s Theorem I: Fourier Analytic Proof over Finite Field
• Lecture 19: Roth’s Theorem II: Fourier Analytic Proof in the Integers
• Lecture 20: Roth’s Theorem III: Polynomial Method and Arithmetic Regularity
• Lecture 21: Structure of Set Addition I: Introduction to Freiman’s Theorem
• Lecture 22: Structure of Set Addition II: Groups of Bounded Exponent and Modeling Lemma
• Lecture 23: Structure of Set Addition III: Bogolyubov’s Lemma and the Geometry of Numbers
• Lecture 24: Structure of Set Addition IV: Proof of Freiman’s Theorem
• Lecture 25: Structure of Set Addition V: Additive Energy and Balog-Szemerédi-Gowers Theorem
• Lecture 26: Sum-Product Problem and Incidence Geometry